Oort's conjecture on direct sums of Newton polygons of curves

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For i=1,2i=1,2, let gig_i be positive integers, let ξi\xi_i be symmetric Newton polygons of height 2gi2g_i, and let g=g1+g2g=g_1+g_2. Define ξ=ξ1⊕ξ2\xi=\xi_1\oplus\xi_2 to be the symmetric Newton polygon of height 2g2g obtained by taking the union of the slopes of ξ1\xi_1 and ξ2\xi_2, with the multiplicity of each slope equal to the sum of its multiplicities in ξ1\xi_1 and ξ2\xi_2.

Oort's conjecture. If ξi\xi_i occurs on Mgi\mathcal{M}_{g_i} for i=1,2i=1,2, then ξ1⊕ξ2\xi_1\oplus\xi_2 occurs on Mg1+g2\mathcal{M}_{g_1+g_2}.

This conjecture predicts that Newton polygons realized by curves can be combined through direct sum. The paper studies some cases of the conjecture, while the general assertion remains open.

References

Primary source

Rachel Pries, “Some cases of Oort's conjecture about Newton polygons”, arXiv:2306.11080 (2024).

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